System and method for nonlinear signal enhancement that bypasses a noisy phase of a signal

ABSTRACT

A system and method for nonlinear signal enhancement is provided. The method comprises: performing a linear transformation on a measured signal comprising a source component and a noise component; determining a modulus of the linear transformed signal; estimating a noise-free part of the linear transformed signal; and reconstructing the source component of the measured signal using the noise-free part of the linear transformed signal.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.60/550,751, filed Mar. 5, 2004, a copy of which is herein incorporatedby reference.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to signal enhancement, and moreparticularly, to a system and method for nonlinear signal enhancementthat does not use a noisy phase of a signal.

2. Discussion of the Related Art

Current signal enhancement techniques are directed to suppressing noiseand improving the perceptual quality of a signal of interest. Forexample, by using signal enhancement algorithms, signal enhancementtechniques can remove unwanted noise and interference found in speechand other audio signals while minimizing degradation to the signal ofinterest. Similarly, image enhancement techniques aim to improve thequality of a picture for viewing. In both cases, however, there is roomfor improvement due to the random nature of noise and the inherentcomplexities involved in speech and signal recognition.

Current signal enhancement techniques follow the approach shown, forexample, in FIG. 1. As shown in FIG. 1, a linear transformation such asa Fourier transform is applied to a noisy signal giving a representationof the signal in the transformed domain (110). The modulus or absolutevalue of the transformed signal is then determined (120) and astatistical estimate of a noise free part of the signal is computed(130). As the statistical estimate is being computed, the phase of thetransformed signal is found (140). The product of the statisticalestimate and the phase of the transformed signal is then determined(150) and an inverse linear transform is applied to the product toinvert the product back into its original domain (160), thus resultingin a cleaned version of the signal.

Such algorithms have been shown to yield significant improvements forlarge classes of signals. However, recent psychoacoustic studies haveshown that signal quality is very dependent on phase estimation. Forexample, if one takes a speech signal, performs a Linear PredictiveCoding analysis and uses random white noise for excitation, thereconstructed signal in the time domain sounds very machine-made. Yet,if one uses custom excitation signals, the signal quality improvesdramatically; however, this technique requires an estimate of the signalphase.

In order to enhance signal quality, information loss that results whentaking the modulus of a signal has been considered. For example, inoptics-based applications, a discrete signal may be reconstructed fromthe modulus of its Fourier transform under constraints in both theoriginal and Fourier domain. For finite signals, the approach uses theFourier transform with redundancy, and all signals having the samemodulus of the Fourier transform satisfy a polynomial factorization.Thus, in one dimension, this factorization has an exponential number ofpossible solutions and in higher dimensions, the factorization is shownto have a unique solution.

Accordingly, there is a need for a technique of accuratelyreconstructing a signal without using its noisy phase or estimation andthat takes into account information loss of the modulus of the signal.

SUMMARY OF THE INVENTION

The present invention overcomes the foregoing and other problemsencountered in the known teachings by providing a system and method fornonlinear signal enhancement that bypasses a noisy phase of a signal.

In one embodiment of the present invention, a method for nonlinearsignal enhancement, comprises: performing a linear transformation on ameasured signal comprising a source component and a noise component;determining a modulus of the linear transformed signal; estimating anoise-free part of the linear transformed signal; and reconstructing thesource component of the measured signal using the noise-free part of thelinear transformed signal.

The step of reconstructing the source component of the measured signal,comprises: performing a nonlinear transformation on the noise-free partof the linear transformed signal; determining a sign of the sourcecomponent of the measured signal; determining a product of the nonlineartransformed signal and the sign; and performing an overlap-add procedureusing the product of the nonlinear transformed signal and the sign.

The linear transformation is one of a Fourier transform and a wavelettransform. The noise-free part of the linear transformed signal isestimated using one of a Wiener filtering technique and an Ephraim-Malahestimation technique.

The noise-free part of the linear transformed signal is estimated bysolving: ${Y\left( {k,\omega} \right)} = \left\{ {\begin{matrix}{{{X\left( {k,\omega} \right)}} - \sqrt{R_{n}\left( {k,\omega} \right)}} & {if} & {{{X\left( {k,\omega} \right)}}^{2} \geq {R_{n}\left( {k,\omega} \right)}} \\0 & {if} & {otherwise}\end{matrix},{{{where}{R_{n}\left( {k,\omega} \right)}} = {\min_{{k - W} \leq k^{\prime} < k}{R_{x}\left( {k^{\prime},\omega} \right)}}},{{{and}{R_{x}\left( {k,\omega} \right)}} = {{\left( {1 - \beta} \right){R_{x}\left( {{k - 1},\omega} \right)}} + {\beta{{{X\left( {k,\omega} \right)}}^{2}.}}}}} \right.$

The step of reconstructing the source component of the measured signalcomprises: defining a three layer neural network by:${q_{k} = {\sigma\left( {{\sum\limits_{f = 1}^{F}{a_{kf}Z_{f}}} + \theta_{k}} \right)}},{1 \leq k \leq L},{and}$${z_{m} = {\sigma\left( {{\sum\limits_{k = 1}^{L}{b_{mk}q_{k}}} + \tau_{m}} \right)}},{{1 \leq m \leq M};}$performing a nonlinear transformation on the noise-free part of thelinear transformed signal by solving:${u_{m} = {z_{m}\sqrt{\frac{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}{z_{1}^{2} + \ldots\quad + z_{M}^{2}}}}};$determining a sign of the source component of the measured signal bysolving: $\rho = \left\{ {\begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {otherwise}\end{matrix};} \right.$determining a product of the nonlinear transformed signal and the sign;and performing an overlap-add procedure using the product of thenonlinear transformed signal and the sign.

The method further comprises iterating:${\pi^{t + 1} = {\pi^{t} - {\alpha\frac{\partial}{\partial\pi}{\sum\limits_{m = 1}^{M}{{u_{m} - s_{m}}}^{2}}}}},$until π converges, wherein π=(A, B, θ, τ). The noise-free part of thelinear transformed signal is estimated by solving:$\min_{{0 \leq a_{k} < {2\pi}},{2 \leq k \leq F}}{\sum\limits_{k = 1}^{F}{\left. {Y_{k}^{\prime} - {{TU}\left( Y^{\prime} \right)}} \right)k{^{2}{,{Y^{\prime} = \left( {{\mathbb{e}}^{{j\alpha}_{k}}Y_{k}} \right)_{1 \leq k \leq F}},{{\alpha_{1} = 0};}}}}}$and the step of reconstructing the source component of the measuredsignal comprises: performing a nonlinear transformation on thenoise-free part of the linear transformed signal by solving:z=U(Y ^(o)), Y _(k) ^(o) =e ^(ja) ^(k) ^(o) Y _(k);determining a sign of the source component of the measured signal bysolving: $\rho = \left\{ {\begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {otherwise}\end{matrix};} \right.$determining a product of the nonlinear transformed signal and the sign;and performing an overlap-add procedure using the product of thenonlinear transformed signal and the sign.

The step of reconstructing the source component of the measured signalcomprises: (i) setting k=0, Y₀=Y; (ii) computing 2 _(k)=UY_(k); (iii)computing W=T_(z) _(k) ; (iv) computing Y₀ using:${{Y_{k + 1}(n)} = {{Y(n)}\frac{W(n)}{{W(n)}}}},{n = 1},2,\ldots\quad,F,{{{wherein}\quad{if}\quad{{Y_{k} - Y_{k + 1}}}} > {ɛ\text{:}}}$incrementing k=k+1, repeating steps (i-iv); and estimating the sourcecomponent of the measured signal using z_(k). The method furthercomprises outputting the reconstructed source component of the measuredsignal.

In another embodiment of the present invention, a system for nonlinearsignal enhancement, comprises: a memory device for storing a program; aprocessor in communication with the memory device, the processoroperative with the program to: perform a linear transformation on ameasured signal comprising a source component and a noise component;determine a modulus of the linear transformed signal; estimate anoise-free part of the linear transformed signal; and reconstruct thesource component of the measured signal using the noise-free part of thelinear transformed signal

When the source component of the measured signal is reconstructed theprocessor is further operative with the program code to: perform anonlinear transformation on the noise-free part of the lineartransformed signal; determine a sign of the source component of themeasured signal; determine a product of the nonlinear transformed signaland the sign; and perform an overlap-add procedure using the product ofthe nonlinear transformed signal and the sign. The measured signal isreceived using one of a microphone and a database comprising one ofaudio signals and image signals.

When the source component of the measured signal is reconstructed theprocessor is further operative with the program code to: define a threelayer neural network by:${q_{k} = {\sigma\left( {{\sum\limits_{f = 1}^{F}{a_{kf}Z_{f}}} + \theta_{k}} \right)}},{1 \leq k \leq L},{and}$${z_{m} = {\sigma\left( {{\sum\limits_{k = 1}^{L}{b_{mk}q_{k}}} + \tau_{m}} \right)}},{{1 \leq m \leq M};}$perform a nonlinear transformation on the noise-free part of the lineartransformed signal by solving:${u_{m} = {z_{m}\sqrt{\frac{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}{z_{1}^{2} + \ldots\quad + z_{M}^{2}}}}};$determine a sign of the source component of the measured signal bysolving: $\rho = \left\{ {\begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {otherwise}\end{matrix};} \right.$determine a product of the nonlinear transformed signal and the sign;and perform an overlap-add procedure using the product of the nonlineartransformed signal and the sign.

The noise-free part of the linear transformed signal is estimated bysolving:$\min_{{0 \leq a_{k} < {2\pi}},{2 \leq k \leq F}}{\sum\limits_{k = 1}^{F}{\left. {Y_{k}^{\prime} - {{TU}\left( Y^{\prime} \right)}} \right)k{^{2}{,{Y^{\prime} = \left( {{\mathbb{e}}^{{j\alpha}_{k}}Y_{k}} \right)_{1 \leq k \leq F}},{{\alpha_{1} = 0};}}}}}$and when the source component of the measured signal is reconstructedthe processor is further operative with the program code to: perform anonlinear transformation on the noise-free part of the lineartransformed signal by solving:z=U(Y ^(o)), Y _(k) ^(o) =e ^(ja) ^(k) ^(o) Y _(k);determine a sign of the source component of the measured signal bysolving: $\rho = \left\{ {\begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {otherwise}\end{matrix};} \right.$determine a product of the nonlinear transformed signal and the sign;and perform an overlap-add procedure using the product of the nonlineartransformed signal and the sign.

When the source component of the measured signal is reconstructed theprocessor is further operative with the program code to: (i) set k=0,Y₀=Y; (ii) compute z_(k)=UY_(k); (iii) compute W=T_(z) _(k) ; (iv)compute Y₀ using:${{Y_{k + 1}(n)} = {{Y(n)}\frac{W(n)}{{W(n)}}}},{n = 1},2,\ldots\quad,F,{{{wherein}\quad{if}\quad{{Y_{k} - Y_{k + 1}}}} > {ɛ\text{:}}}$increment k=k+1, repeat steps (i-iv); and estimate the source componentof the measured signal using Z_(k).

The processor is further operative with the program code to output thereconstructed source component of the measured signal. The reconstructedsource component of the measured signal is output to one of aloudspeaker and an automatic speech recognition system.

In yet another embodiment of the present invention, a method fornonlinear signal enhancement, comprises: receiving a signal comprising asource component and a noise component; performing a lineartransformation on the received signal; determining an absolute value ofthe linear transformed signal; estimating a noise-free part of thelinear transformed signal; performing a nonlinear transformation on thenoise-free part of the linear transformed signal; determining a sign ofthe source component of the received signal; determining a product ofthe nonlinear transformed signal and the sign; and performing anoverlap-add procedure on the product of the nonlinear transformed signaland the sign to form a reconstructed signal of the source component ofthe received signal, wherein the reconstructed signal does not comprisethe noise component of the received signal; and outputting thereconstructed signal. The received signal is one of a speech signal andan image signal.

The foregoing features are of representative embodiments and arepresented to assist in understanding the invention. It should beunderstood that they are not intended to be considered limitations onthe invention as defined by the claims, or limitations on equivalents tothe claims. Therefore, this summary of features should not be considereddispositive in determining equivalents. Additional features of theinvention will become apparent in the following description, from thedrawings and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart illustrating a conventional signal enhancementmethod;

FIG. 2 is a block diagram of a computer system for use with an exemplaryembodiment of the present invention; and

FIG. 3 is a flowchart illustrating a method for nonlinear signalenhancement that bypasses a noisy phase of a signal according to anexemplary embodiment of the present invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

FIG. 2 is a block diagram of a computer system 200 for use with anexemplary embodiment of the present invention. As shown in FIG. 2, thesystem 200 includes, inter alia, a personal computer (PC) 210 connectedto an input device 270 and an output device 280. The PC 210, which maybe a portable or laptop computer or a personal digital assistant (PDA),includes a central processing unit (CPU) 220 and a memory 230. The CPU220 includes a nonlinear signal enhancement module 260 that includes oneor more methods for performing nonlinear signal enhancement that doesnot use a noisy phase or estimate of a signal.

The memory 230 includes a random access memory (RAM) 240 and a read onlymemory (ROM) 250. The memory 230 can also include a database, diskdrive, tape drive or a combination thereof. The RAM 240 functions as adata memory that stores data used during execution of a program in theCPU 220 and is used as a work area. The ROM 250 functions as a programmemory for storing a program executed in the CPU 220. The input device270 is constituted by a keyboard, mouse, microphone or an array ofmicrophones and the output device 280 is constituted by a liquid crystaldisplay (LCD), cathode ray tube (CRT) display, printer or loudspeaker.

Before describing the method of nonlinear signal enhancement accordingto an exemplary embodiment of the present invention, its derivation willbe discussed.

1. Initial Considerations

In formulating the method of nonlinear signal enhancement, an additivemodel given by equation (1) is first considered.x(t)=s(t)+n(t), 0≦t≦T  (1)

As shown in equation (1), x(t) is a measured signal, s(t) is an unknownsource signal and n(t) is a noise signal, all signals are considered ata time t. The signal (x(t))_(0≦t≦T) is “vectorized” into sequencevectors (x(k))_(0≦k≦K), where each x(k) is an M-vector x(k)_(n)=x(kB+n),where 0≦n≦M−1, and B is a time step, which is roughly BK=T. A window gof a size M is applied to the measured signal followed by the Fouriertransform of equation (2): $\begin{matrix}{{{X\left( {k,\omega} \right)} = {\frac{1}{\sqrt{F}}{\sum\limits_{n = 0}^{M - 1}{{\mathbb{e}}^{{- 2}{\pi j\omega}\quad{n/F}}{x(k)}_{n}{g(n)}}}}},{0 \leq \omega \leq {F - 1}}} & (2)\end{matrix}$where the window g defines a linear operator T:R^(M)→C^(F). When F>Mequation (2) may be referred to as being redundant by oversampling inthe frequency domain. An overlap fraction $\frac{M}{b}$represents oversampling in the frequency domain. The redundancy byoversampling in the frequency domain of equation (2) is then considered.

For example, the inversion of the transformation of equation (2), whichis a linear transform, is implemented using an “overlap-add” procedureshown below in equation (3): $\begin{matrix}{{z(t)} = {\frac{1}{\sqrt{F}}{\sum\limits_{k}{\sum\limits_{\omega = 0}^{F - 1}{{\mathbb{e}}^{2{\pi j\omega}\quad{t/F}}{Z\left( {k,\omega} \right)}{\overset{\sim}{g}\left( {t - {kB}} \right)}}}}}} & (3)\end{matrix}$where k ranges over the set of integers so that 0≦t−kB≦M−1, and a “dual”window g, which depends on parameters M, B, are computed to give aperfect reconstruction z=x, when Z=X. The inversion of equation (3)defines a linear operator U:C^(F)→R^(M) and the perfect reconstructioncondition reads UT=I, where I is the identity matrix of a size M.

Conventional noise reduction algorithms use equation (2) as an analysisoperator to represent the measured signal in the time-frequency domain.In the time-frequency domain, the estimation procedure implements anonlinear estimator E(.) of the type shown below in equation (4):Y(k,w)=E(|X(k,w)|)  (4)followed by the inversion of equation (3) with, $\begin{matrix}{{Z\left( {k,\omega} \right)} = {{Y\left( {k,\omega} \right)}\frac{X\left( {k,\omega} \right)}{{X\left( {k,\omega} \right)}}}} & (5)\end{matrix}$

Nonlinear functions of E(.), which are used to implement existingestimation algorithms, require additional information such as thestatistics of the noise component or of the signal component, which areobtained separately. These algorithms use equations (3 and 5) to revertto the time-domain. Given the initial considerations, the method of thepresent invention will now be discussed where the transformation ofequation (2) is inverted using only the absolute values of X(k,w).

2. The Reconstruction Scheme

Starting with, for example, only two samples, x₁, x₂, the following fourlinear transformations shown below in equations (6-9) followed by amodulus are considered:Y ₁ =|x ₁ +x ₂|  (6)Y ₂ =|x ₁ +jx ₂|  (7)Y ₃ =|x ₁−2x ₂|  (8)Y ₄ =|x ₁−2jx ₂|  (9)

Direct computations of equations (6-9) produce the following equations(10 and 11): $\begin{matrix}{x_{1} = {\pm \sqrt{{3Y_{2}^{2}} - Y_{1}^{2} - {\frac{1}{2}Y_{3}^{2}}}}} & (10) \\{x_{2} = \frac{Y_{1}^{2} - Y_{2}^{2}}{2x_{1}}} & (11)\end{matrix}$

for x₁≠0, $\begin{matrix}{x_{1} = 0} & (12) \\{x_{2} = {\pm \sqrt{{\frac{1}{2}Y_{3}^{2}} - Y_{1}^{2}}}} & (13)\end{matrix}$otherwise, equations (12 and 13) are produced. As can be seen, thereremains an ambiguity regarding the signs (e.g., +/− signs). Therefore, astochastic principle can be used to determine whether the sign is + or−. To determine a solution, the set (Y₁, Y₂, Y₃, Y₄) of nonnegativenumbers from equations (6-9) has to satisfy a series of constraintsshown in equations (14 and 15): $\begin{matrix}{{2\left( {Y_{1}^{2} - Y_{2}^{2}} \right)} = {Y_{4}^{2} - Y_{3}^{2}}} & (14) \\{{3Y_{2}^{2}} \geq {Y_{1}^{2} + {\frac{1}{2}Y_{3}^{2}}}} & (15)\end{matrix}$

If, for example, there is a 4-tuple of the set of nonnegative numbers(Y₁, Y₂, Y₃, Y₄) that do not satisfy equations (14 and 15), then thereconstruction scheme may be used to interpolate between admissiblevalues.

The algorithms to be discussed below evolve from the abovereconstruction scheme. In these algorithms, one first considers thegeneral case of an F-vector (Y₁, Y₂ . . . Y_(F)). Then, lineartransformations coming from Parceval frames will be considered, thusleading to a scaling as shown in equation (16). $\begin{matrix}{{\sum\limits_{n = 0}^{M - 1}{{x(k)}_{n}}^{2}} = {\sum\limits_{f = 0}^{F}Y_{f}^{2}}} & (16)\end{matrix}$

Consequently, an inversion will be defined as shown in equation (17),$\begin{matrix}{{Z_{k} = \frac{Y_{k}}{\sqrt{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}}},{1 \leq k \leq F}} & (17)\end{matrix}$in addition, the inversion map will implement a map shown in equation(18): $\begin{matrix}{{{Q\text{:}\quad S^{F - 1}}\bigcap\left( R^{+} \right)^{F}}->S^{M - 1}} & (18)\end{matrix}$between an F−1 dimensional unit sphere with nonnegative entries S^(F−1)∩(R⁺)^(F) and the M−1 dimensional unit sphere S^(M−1).

2.1. The Neural Network Algorithm

In the neural network algorithm, one first considers a 3-layer neuralnetwork defined by equations (19 and 20): $\begin{matrix}{{q_{k} = {\sigma\left( {{\sum\limits_{f = 1}^{F}{a_{kf}Z_{f}}} + \theta_{k}} \right)}},{1 \leq k \leq L}} & (19) \\{{z_{m} = {\sigma\left( {{\sum\limits_{k = 1}^{L}{b_{mk}q_{k}}} + \tau_{m}} \right)}},{1 \leq m \leq M}} & (20)\end{matrix}$where A=(a_(kf))_(1≦k≦L, 1≦f≦F), B=(b_(mk))_(1≦m≦M, 1≦m≦M), andθ=(θ_(k))_(1≦k≦L), τ=(τ_(m))_(1≦m≦M) are network parameters. They may becompactly written as π=(A, B, θ, τ). As shown in equation (20) an inputvector Z=(Z₁, Z₂ . . . Z_(F)) is processed to produce an output vectorZ=(Z₁, z₂ . . . z_(M)). To achieve the mapping shown in equation (18),the network output z =(Z_(m))_(1≦m≦M) has to be normalized to the normof the input vector Y, and then assigned a sign as shown below inequation (21): $\begin{matrix}{u_{m} = {\rho\quad z_{m}\sqrt{\frac{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}{z_{1}^{2} + \ldots\quad + z_{M}^{2}}}}} & (21)\end{matrix}$

The sign ρ is decided based on a maximum likelihood estimation principleand assuming the noise is Gaussian with a variance b², the twolikelihoods are defined in equations (22 and 23): $\begin{matrix}{{p\left( {x = {{{u + n}❘\rho} = {+ 1}}} \right)} \propto {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}^{2}}}} & (22) \\{{p\left( {x = {{{u + n}❘\rho} = {- 1}}} \right)} \propto {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}^{2}}}} & (23)\end{matrix}$

Therefore, the sign is determined by equation (24): $\begin{matrix}{\rho = \left\{ \begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {otherwise}\end{matrix} \right.} & (24)\end{matrix}$

The training of the network, for example, the learning of the parametersof π=(A, B, θ, τ), may be done as shown in equation (25):$\begin{matrix}{\pi^{t + 1} = {\pi^{t} - {\alpha\frac{\partial\quad}{\partial\pi}{\sum\limits_{m = 1}^{M}{{u_{m} - s_{m}}}^{2}}}}} & (25)\end{matrix}$with the learning rate a=10⁻⁸, and Y=|T(s+v)| with T being the linearanalysis operator shown, for example, in equation (2) and(v=(v_(m))_(1≦m≦M), s=(s_(m))_(1≦m≦M)) from a training set made ofspeech and noise signals. For training one can generate random vectorsof M components or use a database of speech and noise signals and divideeach signal into vectors of a size M.

2.2. The Distance Optimization Algorithm

In the distance optimization algorithm, a distance minimizationcriterion may be used. More specifically, let Σ denote the set of allpossible F-nonnegative vectors Y obtained by taking the absolute valueof the linear transformation T in equation (2). Given an F-vector Y ε(R⁺)^(F), which is not necessarily in Σ, the closest element in Σ to Yis found, and then it is nonlinearly inverted as shown below in equation(26). $\begin{matrix}{\hat{Y} = {\arg\quad{\min_{Y^{\prime} \in \Sigma}{\sum\limits_{k = 1}^{F}{{Y_{k} - Y_{k}^{\prime}}}^{2}}}}} & (26)\end{matrix}$

Equation (26) is equivalent to the optimization shown in equation (27):$\begin{matrix}{{{\min_{{0 \leq \alpha_{k} < {2\pi}},{2 \leq k \leq F}}{\sum\limits_{k = 1}^{F}{\left. {Y_{k}^{\prime} - {{TU}\left( Y^{\prime} \right)}} \right)k}}}}^{2},{Y^{\prime} = \left( {{\mathbb{e}}^{{j\alpha}_{k}}Y_{k}} \right)_{1 \leq k \leq F}},{\alpha_{1} = 0}} & (27)\end{matrix}$where a₁=0 fixes the sign ambiguity. This optimization then defines theinverse in equation (28):z=ρU(Y⁰), Y _(k) ⁰ =e ^(ja) ^(k) ^(o) Y _(k)  (28)with a⁰=(a_(k) ⁰)_(1≦k≦F), being the solution of equation (27) and ρ=±1being defined by equation (24). The optimization of equation (27) canthen be performed using a gradient algorithm.

2.3. The Iterative Signal Reconstruction Algorithm

The iterative signal reconstruction algorithm works as follows. Let Y ε(R⁺)^(F) be the vector of, for example, absolute value estimates shownby (Z) in FIG. 3. Next, let T denote the linear transformation as shownby (X) in FIG. 3, and let U denote a left universe. In other words, letU denote another linear transformation that can be used with T toperform a perfect reconstruction, UT=Identity (e.g., T can be a Fouriertransform, and U can be an inverse Fourier transform). Then choose ε=0 astopping threshold, e.g., ε=10 ⁻³. The iterative signal reconstructionalgorithm then iterates the following steps:

(1) Set k=0, Y₀=Y.

(2) Compute z_(k)=UY_(k).

(3) Compute W=T_(z) _(k) .

(4) Compute equation (29) below. $\begin{matrix}{{{Y_{k + 1}(n)} = {{Y(n)}\frac{W(n)}{{W(n)}}}},{n = 1},2,\ldots\quad,F} & (29)\end{matrix}$

(5) ∥Y_(k)−Y_(k+1)∥>ε then increment k=k+1 and go to step 2; otherwisestop.

The estimated or reconstructed signal is indicated by the last computedz_(k).

2.4. Nonlinear Signal Reconstruction

Nonlinear signal reconstruction that does not use a noise component of asignal will now be discussed with reference to FIG. 3. As shown in FIG.3, a linear transformation such as a Fourier transform or a wavelettransform is applied to a signal (x) (shown in equation (1)) comprisinga noise component and an unknown source component giving arepresentation of the signal (X) in the transformed domain (310). Thesignal (x) may be acquired using a microphone or a database comprisingaudio signals and image signals. After the signal (x) is transformed,the modulus or absolute value (Z) of thetransformed signal (X) isdetermined (320). Upon determining the modulus (Z) of the transformedsignal, a statistical estimation (Y) of a noise free part of thetransformed signal is determined (330).

The statistical estimation may be performed by using equation (4),Wiener filtering or by using an Ephraim-Malah estimation technique. Thestatistical estimation may also be performed using a spectralsubtracting technique by solving equations (30), (31) and (32):$\begin{matrix}{{Y\left( {k,\omega} \right)} = \left\{ \begin{matrix}{{{X\left( {k,\omega} \right)}} - \sqrt{R_{n}\left( {k,\omega} \right)}} & {if} & {{{X\left( {k,\omega} \right)}}^{2} \geq {R_{n}\left( {k,\omega} \right)}} \\0 & {if} & {otherwise}\end{matrix} \right.} & (30) \\{{R_{n}\left( {k,\omega} \right)} = {\min_{{k - W} \leq k^{\prime} < k}{R_{x}\left( {k^{\prime},\omega} \right)}}} & (31) \\{{R_{x}\left( {k,\omega} \right)} = {{\left( {1 - \beta} \right){R_{x}\left( {{k - 1},\omega} \right)}} + {\beta{{X\left( {k,\omega} \right)}}^{2}}}} & (32)\end{matrix}$

After performing the statistical estimation (Y) of the noise free partof the transformed signal, the unknown source component of the signal(x) is reconstructed using the statistical estimation (Y) of noise-freepart of the linear transformed signal (340). The unknown sourcecomponent of the signal (x) may be reconstructed by using the neuralnetwork, distance optimization and iterative signal reconstructionalgorithms. In reconstructing the unknown source component using analternative variant of the neural network and distance optimizationalgorithms, a nonlinear transformation or mapping of the statisticalestimation (Y) is performed (350). The nonlinear mapping in the neuralnetwork algorithm may be performed by using equation (33),$\begin{matrix}{u_{m} = {z_{m}\sqrt{\frac{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}{z_{1}^{2} + \ldots\quad + z_{M}^{2}}}}} & (33)\end{matrix}$after defining a three layer neural network using equations (19 and 20).Nonlinear mapping in the distance optimization algorithm may beperformed by using equation (34),z=U(Y ^(o)),Y_(k) ^(o) =e ^(ja) ^(k) ^(o) Y _(k)  (34)

After nonlinear mapping is performed, the sign (ρ) of the sourcecomponent of the signal (x) is determined (360). The sign (ρ) isdetermined by using equation (24). After determining the sign (ρ), theproduct (u/z) of the sign (ρ) and the nonlinear mapped signal (z) isdetermined (370). Upon determining the product (u/z), a conventionaloverlap-add procedure such as that shown in equation (3), is performedon the product (u/z) (380) and a cleaned signal or reconstructed unknownsource signal Ŝ results, which may then be output to a loudspeaker orfurther analyzed by an automatic speech recognition system.

2.5. Proofs

A set of proofs, which show nonlinear mapping is mathematically welldefined, will now be described. This set of proofs is described in thepaper entitled, “On Signal Reconstruction Without Noisy Phase”, RaduBalan, Pete Casazza, Dan Edidin, Dec. 20, 2004, available at(http://front.math.ucdavis.edu/author/Balan-R*), a copy of which isherein incorporated by reference.

By constructing new classes of Parseval frames for a Hilbert space, itwill be shown that new classes of Parseval frames allow for thereconstruction of a signal without using its noisy phase or itsestimation. Frames are redundant systems of vectors in Hilbert spaces.They satisfy the property of perfect reconstruction, in that any vectorof the Hilbert space can be synthesized back from its inner productswith the frame vectors. More precisely, the linear transformation fromthe initial Hilbert space to the space of coefficients obtained bytaking the inner product of a vector with the frame vectors is injectiveand hence admits a left inverse. The follow proofs will show what kindof reconstruction is possible if one only has knowledge of the absolutevalues of the frame coefficients.

First, consider a Hilbert space H with a scalar product <,>. A finite orcountable set of vectors F={f_(i), i ε I} of H is called a frame ifthere are two positive constants A, B>0 such that for every vector x εH, $\begin{matrix}{{A{x}^{2}} \leq {\sum\limits_{i \in I}{\left\langle {x,f_{i}} \right\rangle }^{2}} \leq {B{x}^{2}}} & (2.1)\end{matrix}$

The frame is tight when the constants can be chosen equal to oneanother, A=B. For A=B=1, F is called a Parceval frame. The numbers <x,f_(i)> are called frame coefficients.

To a frame F associate the analysis and synthesis operators defined by:$\begin{matrix}{{{T\text{:}\quad H}->{l^{2}({\mathbb{I}})}},{{T(x)} = \left\{ \left\langle {x,f_{i}} \right\rangle \right\}_{i \in {\mathbb{I}}}}} & (2.2) \\{{{T^{*}\text{:}\quad{l^{2}({\mathbb{I}})}}->H},{{T^{*}(c)} = {\sum\limits_{i \in I}{c_{i}f_{i}}}}} & (2.3)\end{matrix}$which are well defined due to equation (2.1), and are adjacent to oneanother. The range of T in l²(I) is called the range of coefficients.The frame operator defined by S=T*T:H→H is invertible by equation (2.1)and provides the perfect reconstruction formula: $\begin{matrix}{x = {\sum\limits_{i \in {\mathbb{I}}}{\left\langle {x,f_{i}} \right\rangle S^{- 1}f_{i}}}} & (2.4)\end{matrix}$

Consider now, the nonlinear mapping:M _(a) H→l ²(I), M _(a)(x)={|<x,f _(i)>|}_(iεI)  (2.5)obtained by taking the absolute value entry of the analysis operator.Denote by Hr the quotient space H_(r)=H/˜ obtained by identifying twovectors that differ by a constant phase factor: x˜y if there is a scalarc with |c|=1 so that y=cx. For real Hilbert spaces c can only be +1 or−1, and thus H_(r)=H/{±}1}. For complex Hilbert spaces c can be anycomplex number of modulus one, e^(iφ), and then H_(r)=H/T¹, where T¹ isthe complex unit circle. Thus, two vectors of H in the same ray wouldhave the same image through M_(a). Thus the nonlinear mapping M_(a)extends to H_(r) as:M:H _(r→l) ²(I), M({circumflex over (x)})={|< x,f _(i)>|}_(iεI) , xε{circumflex over (x)}  (2.6)

The following description will also center around the injectivity of themap M. For example, when it is injective, M admits a left inverse,meaning that any vector (e.g., signal) in H can be reconstructed up to aconstant phase factor from the modulus of its frame coefficients.

Referring back to the conventional signal enhancement method shown inFIG. 1, the Ephraim-Malah noise reduction method may be used. Let {x(t),t=1, 2 . . . T} be the samples of a speech signal. These samples arefirst transformed into the time-frequency domain through:$\begin{matrix}{{{X\left( {k,\omega} \right)} = {\sum\limits_{t = 0}^{M - 1}{g(t){x\left( {t + {kN}} \right)}{\mathbb{e}}^{{- 2}{\pi\mathbb{i}\omega}\quad\frac{t}{M}}}}},{k = 0},1,\ldots\quad,\frac{T - M}{N},{\omega \in \left\{ {0,1,\ldots\quad,{M - 1}} \right\}}} & (2.7)\end{matrix}$where g is the analysis window, and M, N are respectively the windowsize, and the time step. Next, a nonlinear transformation is applied to|X(k,w)| to produce a minimum mean square error (MMSE) estimate of theshort-time spectral amplitude: $\begin{matrix}{{Y\left( {k,\omega} \right)} = {\frac{\sqrt{\pi}}{2}\frac{\sqrt{\upsilon\left( {k,\omega} \right)}}{\gamma\left( {k,\omega} \right)}{\exp\left( {- \frac{\upsilon\left( {k,\omega} \right)}{2}} \right)}{{{\left( {1 + {\upsilon\left( {k,\omega} \right)}} \right){I_{0}\left( \frac{\upsilon\left( {k,\omega} \right)}{2} \right)}} + {{\upsilon\left( {k,\omega} \right)}{I_{1}\left( \frac{\upsilon\left( {k,\omega} \right)}{2} \right)}}}}{{X\left( {k,\omega} \right)}}}} & (2.8)\end{matrix}$where I₀I₁ are modified Bessel functions of zero and first order, andv(k, w),y(k, w) are estimates of certain signal-to-noise ratios. Thespeech signal windowed Fourier coefficients are estimated by:$\begin{matrix}{{\hat{X}\left( {k,\omega} \right)} = {{Y\left( {k,\omega} \right)}\frac{X\left( {k,\omega} \right)}{{X\left( {k,\omega} \right)}}}} & (2.9)\end{matrix}$and then transformed back into the time domain though an overlap-addprocedure: $\begin{matrix}{{\hat{x}(t)} = {\sum\limits_{k}{\sum\limits_{\omega = 0}^{M - 1}{{\hat{X}\left( {k,\omega} \right)}\quad{\mathbb{e}}^{2\quad\pi\quad{\mathbb{i}}\quad\omega\frac{t - {kN}}{M}}\quad{h\left( {t - {kN}} \right)}}}}} & (2.10)\end{matrix}$where h is the synthesis window. This example illustrates that nonlinearestimation in the representation domain modifies only the amplitude ofthe transformed signal and keeps its noisy phase.

Similarly, in automatic speech recognition systems, given a voice signal{x(t), t =1, 2 . . . T}, the automatic speech recognition system outputsa sequence of recognized phonemes from an alphabet. The voice signal istransformed into the time-frequency domain by the same discrete windowedFourier transform shown in equation (2.7). The real cepstralcoefficients C_(x)(k, w) are defined as the logarithm of the modulus ofX(k, w):C _(x)(k,w)=log(|X(k,w)|)  (2.11)

Two rationales have been discussed for using this object. First, therecorded signal x(t) is a convolution of the voice signal s(t) with thesource-to-microphone (e.g., channel) impulse response h. In the timefrequency domain, the convolution almost becomes a multiplication andthe cepstral coefficients decouple as shown in equation (2.12):C _(x)(k,w)=log(|H(w)|)+C _(S)(k,w)  (2.12)where H(w) is the channel transfer function, and C_(s) is the voicesignal cepstral coefficient. Because the channel function is invariant,by subtracting the time average we obtain:F _(x)(k,w)=C _(x)(k,w)−E[C _(x)(.,w)]=C _(s)(k,w)−E[C_(s)(.,w)]  (2.13)where ε is the time average operator. Thus, F_(x) encodes informationregarding the speech signal alone, independent of the reverberantenvironment. The second rationale for using C_(x) and thus F_(x), isthat phase does not matter in speech recognition. Thus, by taking themodulus in equation (2.11) one does not lose information about themessage Returning to the automatic speech recognition system, thespectral coefficients F_(x) are fed into several hidden Markov models,one hidden Markov model for each phoneme. The outputs of the hiddenMarkov models give an utterance likelihood of a particular phoneme andthe automatic speech recognition system chooses the phoneme with thelargest likelihood. This example also illustrates that the transformeddomain signal has either a secondary role or none whatsoever.

2.5(a). Analysis of M for Real Frames

Consider the case H=R^(N), where the index set I has cardinality M,I={1,2 . . . M}. Then l²(I)˜R^(M). For a frame F={f₁, f₂ . . . f_(M)} ofR^(N), denote T by the analysis operator, $\begin{matrix}\begin{matrix}{\left. {T\text{:}\quad{\mathbb{R}}^{N}}\rightarrow{\mathbb{R}}^{M} \right.,} & {{{T(x)} = {\sum\limits_{k = 1}^{M}{\left\langle {x,f_{k}} \right\rangle\quad e_{k}}}},} & {x \in \hat{x}}\end{matrix} & (2.14)\end{matrix}$where {e₁, e₂ . . . e_(M)} is the canonical basis of R^(M). Let Wdenotethe range of the analysis map TR^(N) that is an N-dimensional subspaceof R^(M). Recall the nonlinear map we are using: $\begin{matrix}\begin{matrix}{\left. {{\mathbb{M}}^{\mathcal{F}}\text{:}\quad{{\mathbb{R}}^{N}/\left\{ {\pm 1} \right\}}}\rightarrow{\mathbb{R}}^{M} \right.,} & {{{\mathbb{M}}^{\mathcal{F}}\left( \hat{x} \right)} = {\sum\limits_{k = 1}^{M}{{\left\langle {x,f_{k}} \right\rangle }\quad e_{k}}}}\end{matrix} & (2.15)\end{matrix}$

When there is no confusion, F will be dropped from the notation.

Two frames {f_(i)}_(iεI) and {g_(i}) _(iεI) are equivalent if there isan invertible operator T on H with T(f_(i))=g_(i), for all iεI. It isknown that two frames are equivalent if their associated analysisoperators have the same range. Next, deduce that M-element frames onR^(N) are parameterized by the fiber bundle F(N, M; R), which is theGL(N, R) bundle over the Grassmanian Gr(N,M).

The analysis will now be reduced to equivalent classes of frames:

Proposition 2.1. For any two frames F and G that have the same range ofcoefficients, M^(F) is injective if MG is injective.

Proof. Any two frames F={f_(k)} and G={g_(k)} that have the same rangeof coefficients are equivalent, e.g., there is an invertible R:R^(N)→R^(N) so that g_(k)=Rf_(k), l≦k≦M. Their associated nonlinear mapsM^(F), and respectively M^(G), satisfy M^(G)(x)=M^(F)(R*x). This showsthat M^(F) is injective if M^(G) is injective. Consequently, theproperty of injectivity of M depends only on the subspace ofcoefficients W in Gr(N,M).

This result shows that for two frames corresponding to two points in thesame fiber F(N, M; R), the injectivity of the their associated nonlinearmaps would jointly hold true or fail. Because of this, one shall assumethat induced topology is based on manifold Gr(N,M) of the fiber bundleF(N, M; R) into the set of M-element frames of R^(N).

If {f_(i}) _(iεI) is a frame with a frame operator S then(S^(½)f_(i)}_(iεI) is a Parseval frame which is equivalent to {f_(i})_(iεI) and called the canonical Parseval frame associated to{f_(i)}_(iεI). Also, {S⁻¹f_(i)}_(iεI) is a frame equivalent to{f_(i)}_(iεI) and is called the canonical dual frame associated to{f_(i)}_(iεI). Proposition 2.1 shows that when the nonlinear map M^(F)is injective then the same property holds for the canonical dual frameand the canonical Parseval frame.

Given S ⊂ {1,2 . . . M} define a map of σ_(s): R^(M)→R^(M) by theequation:σ_(S)(a ₁ , . . . ,a _(M))=((−1)^(S(1)) a ₁, . . . , (−1)^(S(M)) a_(M))  (2.16)

Clearly σ_(s) ²=id and σ_(s)c=−σ_(s) where S^(C) is the complement of S.Let L^(S) denote the |S|-dimensional linear subspace of R^(M) whereL^(S) ={(a₁, a₂ . . . a_(M))|a_(i−)0, i εS}, and let P^(S): R^(M)→L^(S)denote the orthogonal projection onto this subspace. Thus,(P_(S)(u))_(i)=0, if i ε S, and (P_(s)(u))_(i)=u_(i), if i ε S^(C). Forevery vector u ε R^(M), σ₂ (u) =u iff u ε L^(S). Likewise σ_(s)(u)=−uiff u ε (L^(S))^(C). Note: $\begin{matrix}\begin{matrix}{{{P_{S}(u)} = {\frac{1}{2}\left( {u + {\sigma_{S}(u)}} \right)}},} & {{P_{S^{C}}(u)} = {\frac{1}{2}\left( {u - {\sigma_{S}(u)}} \right)}}\end{matrix} & (2.17)\end{matrix}$Theorem 2.2. (Real Frames). If M≧2N−1 then for a generic frame F, M isinjective. Generic means an open dense subset of the set of allM-element frames in R^(N).Proof. Suppose that x and x′ have the same image under M=M_(F). Let a₁,a₂ . . . a_(M) be the frame coefficients of x, and a₁ . . . a_(M) theframe coefficients for x′. Then a_(i)=±a_(i) for each i. In particular,there is a subset S ⊂{1, 2 . . . M} of indices such thata_(i)=(−1)^(S(i))a_(i) where the function S(i) is the characteristicfunction of S and is defined by the rule that S(i)=1 if i ε S and S(i)=0if iε S. Then two vectors x, x′ have the same image under M if there isa subset S⊂{1, 2 . . . M} such that (a₁, a₂ . . . a_(M)) and((−1)^(s(1)) a₁, a₂ . . . (−1)^(S(M)) a_(M)) are both in W the range ofcoefficients associated with F.

To finish the proof it will be shown that when M≧2N −1 such a conditionis not possible for a generic subspace W⊂R^(N). This means that a set ofsuch W's is a dense (e.g., Zariski) open set in the Grassmanian Gr(N,M).In particular, the probability that a randomly chosen W will satisfythis condition is 0.

To finish the proof the following Lemma is needed.

Lemma 2.3. If M≧2N−1 then the following holds for a genericN-dimensional subspace W⊂R^(M). Given u ε W then σ_(s)(u) ε W iffσ_(s)(u)=±u.

Proof. Suppose u ε W and σ_(s)(u)˜±u but σ_(s)(u) εW. Because σ_(s) isan involution, u+σ_(s)(u) is fixed by σ_(s) and is non-zero, thusW∩L^(S)≠0. Likewise,0≠u−σ _(S)(u)=u+σ _(S) c(u)  (2.18)

Therefore, W∩ (L^(S))^(C≠)0.

Now L^(S) and (L^(S))^(C) are fixed linear subspaces of dimension M−|S|and |S|. If M ≧2N−1 then one of these subspaces has a co-dimensiongreater than or equal to N. However a generic linear subspace W ofdimension N has 0 intersections with a fixed linear subspace ofco-dimension greater than or equal to N. Therefore, if W is generic andx, σ_(s)(x) ε W then σ_(s)(x)=±x which ends the proof of the Lemma.

The proof of the theorem now follows from the fact that if W is in theintersection of generic conditions imposed by the proposition for eachsubset S⊂{1, 2 . . . M} , then W satisfies the conclusion of thetheorem.

The proof of Lemma 2.3 actually shows:

Corollary 2.4. The map M is injective if when there is a non-zeroelement u ε W ⊂ R^(M) with u ε L^(S), then W ∩ (L^(S))^(C)={0}.

Now it will be observed that the result is very good.

Proposition 2.5. If M≧2N−2, then the result fails for all M-elementframes.

Proof of Proposition 2.5. Because M≧2N−2, we have that 2M−2N+2≦M. Let(e_(i))_(i=1) ^(M) be the canonical orthonormal basis of R^(M).(e_(i))_(i=1) ^(M)=(e_(i))_(i=1) ^(k)∪ (e_(i))_(i=k+1) ^(M) can then bewritten where both k and M−k are ≧M−N+1.

Let W be any N-dimensional subspace of R^(M). Because dim W^(⊥)=M−N,there exists a nonzero vector u ε (e_(i))_(i=l) ^(k) so that u ⊥ W^(⊥),hence u ε W. Similarly, there is a nonzero vector v in span(e_(i))_(i=k+1) ^(M) with u ⊥ W^(⊥), that is v ε W. By the abovecorollary, M cannot be infective. In fact M(u+v)=M(u−v).

The next result provides an easy way for frames to satisfy the abovecondition.

Corollary 2.6. If F is an M-element frame for R^(N) with M≧2N−1 havingthe property that every N-element subset of the frame is linearlyindependent, then M is infective.

Proof. Given the conditions, it follows that W has no elements, whichare zero in N coordinates, so the Corollary holds.

Corollary 2.7. (1) If M=2N−1, then the condition given in Corollary 2.6is also necessary. (2) If M≧2N, this condition is no longer necessary.

Proof. (1) For (1) in Corollary 2.7, the contrapositive will be proven.Let M=2N−1 and assume there is an N-element subset (f_(i))_(iεS) of Fwhich is not linearly independent. Then there is a nonzero x ε (span(f_(i))_(iεS)) ^(195 ⊂ R) ^(N) Hence, 0≠u=T(x) ε L^(S) ∩ W. On the otherhand, because dim(span(f_(i))_((i εS)) ^(C))≦N−1, there is a nonzero y ε(span(f_(i))_((i εS)) ^(C))^(⊥) ⊂R^(N) so that 0≠v=T(y) ε(L^(S))^(C) ∩W. Now, by Corollary 2.4, M is not injective.

(2) If M≧2N an M-element frame for R^(N) that has a linearly dependentsubset is constructed. Let F′={f₁, . . . f_(2N−1)} be a frame for R^(N)so that any N-element subset is linearly independent. By Corollary 2.4,the map M^(F′) is infective. Now extend this frame to F={f₁ . . . f_(M)}by f_(2n)= . . . =f_(M)=f_(2N−1). The map M^(F) extendsM^(F′ and therefore remains infective, whereas any N-element subset that contains two vectors from {f)_(2N−1), f_(2n) . . . f_(M)} is no longer linearly independent.

It is noted that the above-mentioned frames can be constructed “byhand”. For example, start with an orthonormal basis for R^(N), say(f_(i))_(i=1) ^(N). Assume that sets of vectors (f_(i))_(i=1) ^(M) havebeen constructed such that every subset of N vectors is linearlyindependent. Observe the span of all the (N−1)-element subsets of(f_(i))_(i=1) ^(M). Pick f_(M+1) not in the span of any of thesesubsets. Then (f_(i))_(i=1) ^(M+1) has the property that every N-elementsubset is linearly independent.

Now a slightly different proof of this result will be provided thatgives necessary and sufficient conditions for a frame to have therequired properties.

Theorem 2.8. Let (f_(i))_(i=1) ^(M) be a frame for R^(N). The followingare equivalent:

(1) The map M is injective; and

(2) For every subset S ⊂ {1,2 . . . M}, either (f_(i))_(iεS) spans R^(N)or (f_(i))_((i εS)) ^(C) spans R^(N).

Proof. (1)→(2): The contrapositive is proven. Therefore, assume thatthere is a subset S ⊂ {1, 2 . . . M} so that neither {f_(i); iεS} nor{f_(i)i ε S^(C)} spans R^(N). Hence there are nonzero vectors x, y εR^(N) so that x ⊥ span (f_(i))_(iεS) and y ⊥ span (f_(i))_(i εS) ^(C).Then 0≠T(x) εL^(S) ∩ W and 0≠T(x) ε (L^(S))^(C) ∩W. Now by Corollary 2.4M cannot be injective.

(2)→(1): Assume M({circumflex over (x)}) for some M(ŷ). This means forevery 1≦j≦M, |<x, f_(j)>|=|<y,f_(j>| where x ε {circumflex over (x)} and y ε {circumflex over (y )}. Let,)S={j:<x,f _(j) ]=−y, f _(j]})  (2.19)and note,S ^(C) ={j:[x,f _(j) ]=[y,f _(j)]}  (2.20)

Now, x+y ⊥ span (f_(i))_(iεS) and x−y ⊥ span (f_(i))_(iεS) ^(C). Assumethat {f_(i); i ε S} spans R^(N). Then x+y=0 and thus {circumflex over(x)}=ŷ. If spans {f_(i); i ε S^(C)} then x−y=0 and again {circumflexover (x)}=ŷ. Either way {circumflex over (x)}=ŷwhich proves M isinjective.

For M<2N−1 there are numerous frames for which M is not injective.However for a generic frame, it can be shown that the set of rays thatcan be reconstructed from the image under M is open dense in R^(N)/{±1}.

Theorem 2.9. Assume M>N. Then for a generic frame F εF[N, M; R], the setof vectors x ε R^(N) so that (M^(F))⁻¹ (M_(a) ^(F)(x)) consists of onepoint in R^(N)/{±1} has a dense interior in R^(N).

Proof. Let F be an M-element frame in R^(N). Then F is similar to aframe G that consists of the union of the canonical basis of R^(N) {d₁ .. . d_(N)}, with some other set of M−N vectors. Let G=}(gk; 1≦k≦M}.Thus, gk_(j)=d_(j) 1≦k≦M, for some N elements (k ₁, k₂ . . . k_(N)} of{1, 2 . . . M}. Consider now the set of B frames F so that its similarframe G constructed above has a vector g_(k) with all entries zero,$\begin{matrix}{{\mathbb{B}} = \left\{ {{\left. {\mathcal{F} \in {\mathcal{F}\left\lbrack {N,{M;{\mathbb{R}}}} \right\rbrack}} \middle| {\mathcal{F} \sim \mathcal{G}} \right. = \left\{ {\mathcal{g}}_{k} \right\}},{\left\{ {d_{1},\ldots\quad,d_{N}} \right) \Subset \mathcal{G}},{{\prod\limits_{j = 1}^{N}\left\langle {{\mathcal{g}}_{k_{0}},d_{j}} \right\rangle} \neq 0},{{for}\quad{some}\quad k_{0}}} \right\}} & (2.21)\end{matrix}$

Clearly B is open dense in F[N, M; R]. Thus, generically F ε B. LetG={gk; 1≦k≦M} be its similar frame satisfying the condition above. Next,prove the set X=X^(F) of vectors x ε R^(N) so that (M^(G))⁻¹ (M_(a)^(G)(x)) has more than one point that is thin, e.g., it is included in aset whose complement is open an dense in R^(N). Then claim X ⊂U_(s)(V_(s) ^(+∩ V) _(s) ⁻) where (V_(S) ^(±))_(S⊂{1,2 . . . N}) arelinear subspaces of R^(N) of codimension 1 indexed by subsets S of {1,2. . . N}. This claim will conclude the proof of Theorem 2.9.

To verify the claim, let x, y, ε R^(N) so that M_(a) ^(G) (x)=M_(a) ^(G)(y) and yet x≠y, nor x≠−y. Because G contains the canonical basis ofR^(N), |x_(k)|=|y_(k)| for all 1 ≦k≦N. Then there is a subset S ⊂ {1,2 .. . N{ so that y_(k)=(−1)^(S(k))x_(k). Note S≠0, nor S≠0, nor S≠{1, 2 .. . N}. Denote D_(S) the diagonal N×M matrix (D_(S))_(kk)=(−1)^(S(k)).Thus y=D_(S) ^(x) and yet D_(S≠±)1Let G_(k0) ε G so that none of itsentries vanish. Then |<x, g_(k0)>|=|<y, g_(ko)>| implies,[x, (I±D _(S))g _(ko)]=0  (2.22)

This proves the set X^(G) is included into the union of 2(2^(N)−2)linear subspaces of codimension 1,∪_(S≠0,S) C _(≠0){(I−D _(S))gk _(o)}⊥∪{(+D _(S))gk_(o)}^(⊥)  (2.23)

Because F is similar to G, X^(F) is included into the image of the aboveset through a linear invertible map, which proves the claim.

2.5(b). Analysis for M Complex Frames

In this section the Hilbert space is C^(N). For an M-element frameF={f₁, f₂ . . .f_(M)} of C^(N) the analysis operator is defined byequation (2.14), where the scalar product is <x,$\left\langle {x,y} \right\rangle = {\sum\limits_{k = 1}^{N}{{x(k)}\quad{\overset{\_}{y(k)}.}}}$The range of coefficients, e.g., the range of the analysis operator, isa complex N-dimensional subspace of C^(M) that is denoted again by W.The nonlinear map to be analyzed is given by: $\begin{matrix}\begin{matrix}{\left. {{\mathbb{M}}^{\mathcal{F}}\text{:}\quad{{\mathbb{C}}^{N}/{\mathbb{T}}^{1}}}\rightarrow{\mathbb{C}}^{M} \right.,} & {{{{\mathbb{M}}^{\mathcal{F}}\left( \hat{x} \right)} = {\sum\limits_{k = 1}^{M}{{\left\langle {x,f_{k}} \right\rangle }\quad e_{k}}}},} & {x \in \hat{x}}\end{matrix} & (2.24)\end{matrix}$where two vectors x, y ε {circumflex over (x)} if there is a scalar c εC with |c|−1 so that y=cx.

By the equivalence, M-frames of C^(N) are parameterized by points of thefiber bundle F(N, M; C), and the GL(N, C) bundle over the complexGrassmanian Gr(N, M)^(C).

Proposition 2.1 holds true for complex frames as well. Thus without lossof generality the topology induced by the base manifold of F(N, M; C)into the set of M-element frames of C^(N) will be used. Thus, in thereal case, the question about M-element frames in C^(N) is reduced to aquestion about the Grassmanian of N-planes in C^(M). First, thefollowing theorem is proved.

Theorem 3.1. If M≧4N−2 then the generic N-plane W in C^(M) has theproperty that if v =(v₁, v₂ . . . v_(M)) and w=(w₁, w₂ w_(M)) arevectors in W such that |v_(i)=|w_(i)| for all i then vλw for somecomplex number λ of modulus 1.

Proof. Assume that an N-plane W has a property (*) if there arenon-parallel vectors v, w in W such that |v_(i)|=|w_(i)| for all i.Recall that two vectors x, y are parallel if there is a scalar c ε C sothat y=cx.

Given an N-plane W it may be assumed, after recording coordinates onC^(M), that W is the span of the rows of an N x M matrix of the form:$\begin{matrix}\begin{bmatrix}1 & 0 & \ldots & 0 & u_{{N + 1},1} & \ldots & u_{M,1} \\0 & 1 & \ldots & 0 & u_{{N + 1},2} & \ldots & u_{M,2} \\❘ & ❘ & ❘ & ❘ & ❘ & ❘ & \quad \\0 & 0 & \ldots & 1 & u_{{N + 1},N} & \ldots & u_{M,N}\end{bmatrix} & (2.25)\end{matrix}$where the N (M−N) entries {U_(i,j)} are viewed as indeterminates. Thus,Gr(N, M)^(C) is isomorphic to C^(N(M−N)) in a neighborhood of W.

Now suppose that W satisfies (*) and v and w are two non-parallelvectors whose entries have the same modulus. The choice of basis for Wensures that one of the first N entries in v (and hence w) are nonzero.Because one only cares about these vectors up to rescaling, it may beassumed, after recording, that v₁=w₁=1. In addition, the vectors areassumed non-parallel so that it may be assumed that v₁≠w₁≠0 for somei≦N. After again reordering, it can be assumed that v₂≠w₂≠0.

Then set λ₁=1. By assumption there are numbers λ₂ . . . λ_(M) with λ₂161 such that w_(i)=λ_(i)v_(i) 1, 2 . . . M. Expanding in terms of thebasis for W, for i> ${v_{i} = {\sum\limits_{j = 1}^{N}{v_{j}u_{i,j}}}},$and $w_{i} = {\sum\limits_{j = 1}^{N}{\lambda_{j}v_{j}{u_{i,j}.}}}$

Thus if W satisfies (*) there must be λ₂ . . . λ_(N) ε T¹ (with λ₁≠1)and v₂ . . . v_(N) ε C such that for all N+1≦i≦M one has,$\begin{matrix}{{{\sum\limits_{j = 1}^{N}{v_{j}u_{i,j}}}} = {{\sum\limits_{j = 1}^{N}{\lambda_{j}v_{j}u_{i,j}}}}} & (2.26)\end{matrix}$

Consider now, the variety Y of all tuples,(W, u ₃ , . . . ,u _(N), λ₂, . . . , λ_(N))  (2.27)as above. Because v₂≠0 and λ₂≢1 this variety is locally isomorphic tothe real 2N(M −N)+3N−3-dimensional variety,C ^(N(M−N))×(C\{0})×(C)^(N−2)×(T ¹\{1})×(T ¹)^(N−2)  (2.28)

The locus in Gr(N, M)^(C) of planes satisfying the property (*) isdenoted by X. This variety is the image under projection to the firstfactor of Y cut out by the M−N equation (2.26) for N+1≦i≦M. The analysisof equation (2.26) is summarized by the following result.

Lemma 3.2. The M−N equations of (2.26) are independent. Hence X is avariety of a real dimension at most 2N(M−N)+3N−3−(M−N).

Proof of Lemma 3.2. For any choice of 0≠v₂, v₃ . . . v_(N) and 1≠λ₂, λ₃. . . λ_(N) the equation, $\begin{matrix}{{{\sum\limits_{j = 1}^{M}{\upsilon_{j}u_{i,j}}}}^{2} = {{\sum\limits_{j = 1}^{M}{\lambda_{j}\upsilon_{j}u_{i,j}}}}^{2}} & (2.29)\end{matrix}$is non-degenerate. Because the variables u_(i,1) . . . u_(i,N) appear inexactly one equation, these equations (for fixed v₂, v₃ . . . v_(N), λ₂,λ₃ . . . λ_(N)) define a subspace of C^(N(M−N)) of a real codimension atleast M−N. Because this is true for all choices, it follows that theequations are independent.

From this Lemma, it follows that the locus of N-planes satisfying (*)has a local or real dimension 2N(M−N)+3N−3−(M−N). Therefore, if3N−3−(M−N)<0, e.g., if M≧4N−2, this locus cannot be all of Gr(N, M)^(C),thus ending the proof of Theorem 3.1.

The main result in the complex case then follows from Theorem 3.1.

Theorem 3.3 (Complex Frames). If M≧4N−2 then M^(F) is injective for ageneric frame F={f₁, f₂ . . . f_(N)}.

Lemma 3.2 yields the following result.

Theorem 3.4. If M≧2N then for a generic frame F ε F|N, M; C| the set ofvectors x ε C^(N) such that (M^(F))⁻¹(M_(a) ^(F)(x)) has one point inC^(N) |T¹ has a dense interior in C^(N).

Proof. From Lemma 3.2, for a generic frame the M−N equations of (2.26)in 2(N−1) indeterminates (v₂ . . . v_(N), λ₂ . . . λ_(N)) areindependent. Note there are 3(N−1) real valued unknowns and M−Nequations. Hence, the set of {(v₂ . . . v_(N)} in C^(N−1) for whichthere are (λ₂ . . . λ_(N)) such that equation (2.26) has a solution in(C\{0})×(C)^(N−2)×(T¹ 544 {1})×(T¹)^(N−2) has a real dimension at most3(N31 1)−(M−N)=4N−3−M. For M≧2N it follows 3(N−1)−(M−N)<2(N−1) whichshows the set of v=(v₁ . . . v_(N)) such that (M^(F))⁻¹(M_(a) ^(F)(v))has more than one point is thin in C^(N), e.g., its complement has adense interior.

The optimal bound for the complex case is thus believed to be 4N−2.However, as this case is different from the real case in that complexframes with only 2N−1 elements cannot have M^(F) injective. To showthis, the proof of Theorem 2.8 (1)→(2) does not use the fact that theframes are real. So in the complex case, one has: Proposition 3.5. If{f_(j)}_(jε I) is a complex frame and M^(F) is injective, then for everyS ⊂ {1,2 . . . M}, if L^(S) ∩ W≠0 then (L^(S))^(C) ∩ W. Hence, for everysuch S, either {f_(j)}_(jεs) or (f_(j))_(jεS) ^(C) spans H.

Now it will be shown that complex frames must contain at least2N-elements for M^(F) to be injective.

Proposition 3.6 (Complex Frames). If M^(F) is injective then M≧2N.

Proof. It is assumed that M=2N−1 and that in this case M^(F) is notinjective. Let {Z_(j)}_(j=1) ^(N) be a basis for W and let P be theorthogonal projection onto the first N−1 unit vectors in C^(M). Then{Pz_(j)}_(j=1) ^(N) sits in an N−1-dimensional space and so there arecomplex scalars {a_(j)}_(j = 1)^(N − 1),not all zero, so that Σa_(j)P_(Zj)=0. In other words, there is a vector0≠y ε with support y ⊂{N, N+1 . . . 2N−1}. Similarly, there is a vector0≠y ε with support x ⊂{1,2 . . . N}. If x(N)=0 or y(N)=0 we contradictProposition 3.4. In addition, if${{x(i)} = {{0\quad{for}\quad{all}\quad i} < N}},{{{then}\quad\left( {y - {cx}} \right)(N)} = {{0\quad{for}\quad c} = {{y(N)}{\frac{\overset{\_}{x(N)}}{{{x(N)}}^{2}}.}}}}$Now, x, y−cx are in W and have disjoint support so the map is notinjective. Otherwise, let, $\begin{matrix}\begin{matrix}{{z = \frac{\overset{\_}{x(N)}}{{{x(N)}}^{2}}},} & {w = {i\frac{\overset{\_}{y(N)}}{{{y(N)}}^{2}}}}\end{matrix} & (2.30)\end{matrix}$

Now, z, w ε W and z(N)=1 and w(N)=i. Hence, |z+w|=|z−w|. It follows thatthere is a complex number |c|=1 so that z+w=(c(z−w). Because zi≠0 forsome i<N, c=1 and w=0, which is a contradiction.

Thus, in accordance with an exemplary embodiment of the presentinvention, by constructing new classes of Parseval frames for a Hilbertspace, an original signal that includes an unknown source component anda noise component is reconstructed without using its noise component orestimation. Therefore, by using information available from thetransformed domain of a signal, signal reconstruction may take placewithout using an estimate of signal phase.

It is to be further understood that because some of the constituentsystem components and method steps depicted in the accompanying figuresmay be implemented in software, the actual connections between thesystem components (or the process steps) may differ depending on themanner in which the present invention is programmed. Given the teachingsof the present invention provided herein, one of ordinary skill in theart will be able to contemplate these and similar implementations orconfigurations of the present invention.

It should also be understood that the above description is onlyrepresentative of illustrative embodiments. For the convenience of thereader, the above description has focused on a representative sample ofpossible embodiments, a sample that is illustrative of the principles ofthe invention. The description has not attempted to exhaustivelyenumerate all possible variations. That alternative embodiments may nothave been presented for a specific portion of the invention, or thatfurther undescribed alternatives may be available for a portion, is notto be considered a disclaimer of those alternate embodiments. Otherapplications and embodiments can be implemented without departing fromthe spirit and scope of the present invention.

It is therefore intended, that the invention not be limited to thespecifically described embodiments, because numerous permutations andcombinations of the above and implementations involving non-inventivesubstitutions for the above can be created, but the invention is to bedefined in accordance with the claims that follow. It can be appreciatedthat many of those undescribed embodiments are within the literal scopeof the following claims, and that others are equivalent.

1. A method for nonlinear signal enhancement, comprising: performing alinear transformation on a measured signal comprising a source componentand a noise component; determining a modulus of the linear transformedsignal; estimating a noise-free part of the linear transformed signal;and reconstructing the source component of the measured signal using thenoise-free part of the linear transformed signal.
 2. The method of claim1, wherein the step of reconstructing the source component of themeasured signal, comprises: performing a nonlinear transformation on thenoise-free part of the linear transformed signal; determining a sign ofthe source component of the measured signal; determining a product ofthe nonlinear transformed signal and the sign; and performing anoverlap-add procedure using the product of the nonlinear transformedsignal and the sign.
 3. The method of claim 1, wherein the lineartransformation is one of a Fourier transform and a wavelet transform. 4.The method of claim 1, wherein the noise-free part of the lineartransformed signal is estimated using one of a Wiener filteringtechnique and an Ephraim-Malah estimation technique.
 5. The method ofclaim 1, wherein the noise-free part of the linear transformed signal isestimated by solving: $\begin{matrix}{{Y\left( {k,\omega} \right)} = \left\{ {\begin{matrix}{{{X\left( {k,\omega} \right)}} - \sqrt{R_{n}\left( {k,\omega} \right)}} & {{{if}\quad{{X\left( {k,\omega} \right)}}^{2}} \geq {R_{n}\left( {k,\omega} \right)}} \\0 & {{if}\quad{otherwise}}\end{matrix},{where}} \right.} \\{{R_{n}\left( {k,\omega} \right)} = {\min_{{k - W} \leq k^{\prime} < k}{{R_{x}\left( {k^{\prime},\omega} \right)}\quad{and}}}} \\{{R_{x}\left( {k,\omega} \right)} = {{\left( {1 - \beta} \right)\quad{R_{x}\left( {{k - 1},\omega} \right)}} + {\beta{{{X\left( {k,\omega} \right)}}^{2}.}}}}\end{matrix}$
 6. The method of claim 1, wherein the step ofreconstructing the source component of the measured signal, comprises:p1 defining a three layer neural network by: $\begin{matrix}\begin{matrix}{{q_{k} = {\sigma\left( {{\sum\limits_{f = 1}^{F}{a_{k\quad f}\quad Z_{f}}} + \theta_{k}} \right)}},} & {1 \leq k \leq L} & {and}\end{matrix} \\\begin{matrix}{{z_{m} = {\sigma\left( {{\sum\limits_{k = 1}^{L}{b_{mk}q_{k}}} + \tau_{m}} \right)}},} & {{1 \leq m \leq M};}\end{matrix}\end{matrix}$ performing a nonlinear transformation on the noise-freepart of the linear transformed signal by solving:${u_{m} = {z_{m}\sqrt{\frac{Y_{1}^{2} + \cdots + Y_{F}^{2}}{z_{1}^{2} + \cdots + z_{M}^{2}}}}};$determining a sign of the source component of the measured signal bysolving: $\rho = \left\{ {\begin{matrix}{+ 1} & {{{if}\quad{\sum\limits_{k = 1}^{M}{{x_{k} - {u_{k}}}}^{2}}} \leq {\sum\limits_{k = 1}^{M}{{x_{k} + {u_{k}}}}^{2}}} \\{- 1} & {{if}\quad{otherwise}}\end{matrix};} \right.$ determining a product of the nonlineartransformed signal and the sign; and performing an overlap-add procedureusing the product of the nonlinear transformed signal and the sign. 7.The method of claim 6, further comprising: iterating:$\pi^{t + 1} = {\pi^{t} - {\alpha\quad\frac{\partial}{\partial\pi}{\sum\limits_{m = 1}^{M}{{u_{m} - s_{m}}}^{2}}}}$until π converges, wherein π=(A, B, θ, τ).
 8. The method of claim 1,wherein the noise-free part of the linear transformed signal isestimated by solving:${{\min_{{0 \leq \alpha_{k} < {2\pi}},{2 \leq k \leq F}}{\sum\limits_{k = 1}^{F}{\left. {Y_{k}^{\prime} - {{TU}\left( Y^{\prime} \right)}} \right)k}}}}^{2},{Y^{\prime} = \left( {{\mathbb{e}}^{{j\alpha}_{k}}Y_{k}} \right)_{1 \leq k \leq F}},{{\alpha_{1} = 0};}$and the step of reconstructing the source component of the measuredsignal, comprises: performing a nonlinear transformation on thenoise-free part of the linear transformed signal by solving:z=U(Y ^(o)), Y _(k) ^(o) =e ^(ja) ^(k) ^(o) Y _(k); determining a signof the source component of the measured signal by solving:$\rho = \left\{ \begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {{otherwise};}\end{matrix} \right.$ determining a product of the nonlinear transformedsignal and the sign; and performing an overlap-add procedure using theproduct of the nonlinear transformed signal and the sign.
 9. The methodof claim 1, wherein the step of reconstructing the source component ofthe measured signal, comprises: (i) setting k=0, Y₀=Y; (ii) computingz_(k)=UY_(k); (iii) computing W =Tz_(k); (iv) computing Y₀ using:${{Y_{k + 1}(n)} = {{Y(n)}\frac{W(n)}{{W(n)}}}},{n = 1},2,\ldots\quad,F,$wherein if ∥Y_(k)−Y_(k+1)∥>ε: incrementing k=k+1, repeating steps(i-iv); and estimating the source component of the measured signal usingZ_(k).
 10. The method of claim 1, further comprising: outputting thereconstructed source component of the measured signal.
 11. A system fornonlinear signal enhancement, comprising: a memory device for storing aprogram; a processor in communication with the memory device, theprocessor operative with the program to: perform a linear transformationon a measured signal comprising a source component and a noisecomponent; determine a modulus of the linear transformed signal;estimate a noise-free part of the linear transformed signal; andreconstruct the source component of the measured signal using thenoise-free part of the linear transformed signal
 12. The system of claim11, wherein when the source component of the measured signal isreconstructed the processor is further operative with the program codeto: perform a nonlinear transformation on the noise-free part of thelinear transformed signal; determine a sign of the source component ofthe measured signal; determine a product of the nonlinear transformedsignal and the sign; and perform an overlap-add procedure using theproduct of the nonlinear transformed signal and the sign.
 13. The systemof claim 11, wherein the measured signal is received using one of amicrophone and a database comprising one of audio signals and imagesignals.
 14. The method of claim 11, wherein when the source componentof the measured signal is reconstructed the processor is furtheroperative with the program code to: define a three layer neural networkby:${q_{k} = {\sigma\left( {{\sum\limits_{f = 1}^{F}{a_{kf}Z_{f}}} + \theta_{k}} \right)}},{1 \leq k \leq {L\quad{and}}}$${z_{m} = {\sigma\left( {{\sum\limits_{k = 1}^{L}{b_{mk}q_{k}}} + \tau_{m}} \right)}},{{1 \leq m \leq M};}$perform a nonlinear transformation on the noise-free part of the lineartransformed signal by solving:${u_{m} = {z_{m}\sqrt{\frac{Y_{1}^{2} + \ldots\quad + Y_{F}^{2}}{z_{1}^{2} + \ldots\quad + z_{M}^{2}}}}};$determine a sign of the source component of the measured signal bysolving: $\rho = \left\{ \begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {{otherwise};}\end{matrix} \right.$ determine a product of the nonlinear transformedsignal and the sign; and perform an overlap-add procedure using theproduct of the nonlinear transformed signal and the sign.
 15. The methodof claim 11, wherein the noise-free part of the linear transformedsignal is estimated by solving:${{\min_{{0 \leq \alpha_{k} < {2\pi}},{2 \leq k \leq F}}{\sum\limits_{k = 1}^{F}{\left. {Y_{k}^{\prime} - {{TU}\left( Y^{\prime} \right)}} \right)k}}}}^{2},{Y^{\prime} = \left( {{\mathbb{e}}^{{j\alpha}_{k}}Y_{k}} \right)_{1 \leq k \leq F}},{{\alpha_{1} = 0};{and}}$when the source component of the measured signal is reconstructed theprocessor is further operative with the program code to: perform anonlinear transformation on the noise-free part of the lineartransformed signal by solving:z=U(Y ^(o)), Y _(k) ^(o) =e ^(ja) ^(k) ^(o) Y _(k); determine a sign ofthe source component of the measured signal by solving:$\rho = \left\{ \begin{matrix}{+ 1} & {if} & {\sum\limits_{k = 1}^{M}{{{x_{k} -}}u_{k}{^{2}{\leq {\sum\limits_{k = 1}^{M}{{{x_{k} +}}u_{k}}}}}^{2}}} \\{- 1} & {if} & {{otherwise};}\end{matrix} \right.$ determine a product of the nonlinear transformedsignal and the sign; and perform an overlap-add procedure using theproduct of the nonlinear transformed signal and the sign.
 16. The methodof claim 11, wherein when the source component of the measured signal isreconstructed the processor is further operative with the program codeto: (i) set k=0, Y₀=Y; (ii) compute z_(k)=UY_(k); (iii) computeW=Tz_(k); (iv) compute Y₀ using:${{Y_{k + 1}(n)} = {{Y(n)}\frac{W(n)}{{W(n)}}}},{n = 1},2,\ldots\quad,F,$wherein if ∥Y_(k)−Y_(k+1)∥>ε: increment k=k+1, repeat steps (i-iv); andestimate the source component of the measured signal using z_(k). 17.The system of claim 11, wherein the processor is further operative withthe program code to: output the reconstructed source component of themeasured signal.
 18. The system of claim 17, wherein the reconstructedsource component of the measured signal is output to one of aloudspeaker and an automatic speech recognition system.
 19. A method fornonlinear signal enhancement, comprising: receiving a signal comprisinga source component and a noise component; performing a lineartransformation on the received signal; determining an absolute value ofthe linear transformed signal; estimating a noise-free part of thelinear transformed signal; performing a nonlinear transformation on thenoise-free part of the linear transformed signal; determining a sign ofthe source component of the received signal; determining a product ofthe nonlinear transformed signal and the sign; and performing anoverlap-add procedure on the product of the nonlinear transformed signaland the sign to form a reconstructed signal of the source component ofthe received signal, wherein the reconstructed signal does not comprisethe noise component of the received signal; and outputting thereconstructed signal.
 20. The method of claim 19, wherein the receivedsignal is one of a speech signal and an image signal.